Pumping system analogue



Dec. 8, 1953 F. w. BUBB ,8

' PUMPING SYSTEM ANALOGUE Filed Jan. 8, 1951 I 6 Sheets-Sheet l WQWAQWWA v IF/ 6.

INVENTOR.

- F. W. BUBB HMLW $2? Dc. s, 1953 F. w. BUBB 2,661,898

PUMPING SYSTEM ANALOGUE Filed Jan. 8, 1951' I g 88 I 6Sheets-Sheet 2 FIG. 6.

INVENTOR. F. w. BUBB HMLMX-7 TTOR EYS Dec; 8, 1953 PUMPING SYSTEM Filed Jan. 8, 1951 F. W. BUBB ANALOGUE 6 Sheets-Sheet 3 INVENTOR. F. W. BUBB AT ORNE Filed Jan. 8, 1951 6 Sheets-Sheet 4 Ivy Ir co 0 l; 129 i i cos e r a INVENTOR.

14f Jig 142 F. w. BUBB 14o UNIT 6 4 BY FIG. /7 HM/M f ATTZEVZ Dec. 8, 1953 w, B 2,661,898

PUMPING SYSTEM ANALOGUE Filed Jan. 8, 1951 6 Sheets-Sheet 5 UNlTl o I40 FIG. /8.

' INVENTOR.

( F.W.,BUBB

F/G. 2/. BY

I 70/? rs Dec. 8, 1953 F. w. BUBB 2,661,898

PUMPING SYSTEM ANALOGUE 6 Sheets-Sheet 6 Filed Jan. 8, 1951 Ma B m U VB mw F 'HAJM FIG. 24.

Patented Dec. 8, 1953 PUMPING SYSTEM ANALOGUE Frank Bubb. Webster Groves, Mo., assignor t Phillips Petroleum Company, a corporation of Delaware Application January 8,1951, Serial No. 204,926

This invention relates to a simulator which is an electrical analogue of a pumping system. In one specific aspect, it relates to a simulator which is an electrical analogue of the tubing, sucker rod,

oil column, and pump in a deep well pumping unit. In another specific aspect, it relates to a simulator which is an electrical analogue of the mechanism connecting a prime mover with the top or polished rod of a sucker rod pumping unit.

A typical deep well system includes a series or string of metal'rods, referred to in the art as sucker rods, these rods extending through tubing positioned in the well, the lower end of the tubing carrying a pumping unit. The'tubing, in turn, is suspended within a well casing, and, under some conditions, the lower portion of the interspace between the tubing and easing may contain a column of oil. At the top of the well, the sucker rod string extends through a stufiing box and is driven by a prime mover, such as an electric motor or an internal combustion engine, through a flywheel or bull wheel driven by the prime mover. This flywheel, in turn, drives a crank or 'counter balance which is coupled through a walking beam to the top end of the sucker rod string.

As the pumping operation is carried out, a colstring. The weight and elasticity of the sucker rod string, as well as the tubing and oil column, produce elastic strains in the pumping unit so that the sucker rod string and tubing behave in a manner somewhat analogous to elongated springs.

This elastic movement of the sucker rod string and tubing oftentimes results in failure of one or more sucker rods and, more often, in inefficient operation of the pumping system. It has not been possible, in the past, to predict the effect of changes in various operating variables of the system with any degree of accuracy, except by purely empirical or cut and try methods, for the obvious reason that the movement of the sucker rod string several miles below the "surface'of the earth cannot be observed nor is it possible to observe the manner of operation or" the downhole pump.

It has been proposed to provide a system of weights and springs to behave in the same manner as the pumping system but the difficulties of adjustment of various parts of the system and the ia-claims, (crass- 6 1 It has also been proposed to construct an electrical system in whichspecified variables of the electrical system correspond to certain variables of the pumping system so that the electrical system is an analogueof the actual pumping system.

It has, however, been assumed in such proposals that the oil couldbe subjected both to tension and compression whereas, in fact, it is not possible to place an oil column under tension. Accordingly, the results of such proposed systems do not fully represent the behavior of the actual pumping system. Moreover, such proposals have neglected the efiect of the downhole apparatus upon the driving system which again leads to considerable difiiculty for the downhole system has a definite reaction upon the driving system which cannot be neglected if accurate results are to be obtained.

It is an object of the invention to provide an improved electrical system Which is an analogue of an actual well pumping unit, which system may be utilized to predict the effect of changing various operating conditions of the pumping system.

It is a further object to provide apparatus for simulating thebehaviorof the driving mechanism interposed between "the prime mover and the sucker rod string.

It is a still further object to provide apparatus which is reliable in operation, which utilizes standard circuit components, and is of the lowest cost commensurate with the complexity of the data obtained therefrom.

Various other objects, advantages and features "of, the invention will become apparent from the 55 following detailed description taken in conjunction with the accompanying drawings, in which: Figure 1 is a vertical sectional view, partially in elevation, of a typical deep well pumping system; Figure 2 is a schematic circuit diagram of an electrical simulator for the downhole pumping apparatus;

Figures 3, 4 and 5 are views explaining the derivation of the differential equations of motion of sucker rod, tubing and oil column.

Figures 6 and '7 are schematic circuit diagrams Qofthe lower circuit element of Figure 2 during the upstroke and downstroke periods.

Figure 8 is a schematic circuit diagram of an amplifier system simulating an ideal transformer.

Figures 9 and 10 are schematic views of portions of the drive mechanism in its original position-and in a general position, respectively. Figure 11 is a block diagram of the simulator for the drive mechanism;

- Figures 12 to 20, inclusive, are schematic circuit diagrams of circuits for producing voltages corresponding to the differential equations of motion of various parts of the drive mechanism; and

Figures 21 to 25, inclusive, are schematic diagrams of circuits for coupling the circuit equation units of Figures 12 to 20, inclusive.

Referring now to Figure 1, Ihave illustrated; in a schematic manner, atypical deep well pumping system to which the simulator of this invention is applicable, this system including driving mechanism H] located at the surface and a downhole pumping unit Ii. The unit ll includes the usual well casing :2 within which is mounted tubing it having a sucker rod string l4 mounted therein. The string It includes a number of sucker rods l5 connected together by joints 16, the upper endof the string extending through a stuifing' box 11 in a casinghead l3 and the lower end being attached to a plunger is forming a part of a pump 283 which includes an upper or traveling valve 2! and a bottom or stationary valve 22. The uppermost sucker rod or polished rod 23 is driven by the mechanism it which includes a walking beam 25 having one end secured to the polished rod and its other end secured to a pitman 25 which is driven by a crank or counter-balance 26. The

, crank is driven by a flywheel 21, as through a chain drive 28, and the flywheel, in turn, is' driven by a prime mover 29, as by a second chain drive 3 9. The prime mover "2's may be an electric motor, an internal combustion engine, or any other suitable type of engine.

In accordance with the invention, th operation of the downhole unit H and the drive mechanism I0 is electrically simulated by the apparatus of my invention. The apparatus of Figure 2 will be described first and this apparatus simulates the behavior of the downhole unit II. The voltage at various parts of the electrical system represents stress at corresponding parts of the pumping unit, the charge at various points of the electrical system represents displacement of corresponding parts of the actual pumping system, and current in the electrical system represents the velocity of corresponding parts of the mechanical system. In order for the two systems to be analogues of each other, it is necessary that the differential equations describing the behavior of the electrical system be of similar form to the differential equations describing the behavior of the actual pumping unit. Thereupon, by suitably adjusting the coefficients of various terms of the electrical system, as by changing the parameters of various parts of the electrical system, a condition is obtained whereby the electrical potentials, charges, voltages and other parameters of the electrical system correspond to the stress, displacement, velocity and other corresponding variables of the pumping unit.

At this time, the two systems are analogues of each other and further changes in the parameters of the electrical system produce efiects thereon which enable accurate prediction of the effects of corresponding charges in the pumping unit.

In the simulator, an electrical line or network T represents the tubing l3, an electrical line 0 represents the oil column and an electrical line R represents the sucker rod string [5. The subscripts of the symbols represent the line of the circuit to which they are applicable such as the tubing, oil or rod line. The following notation is used in the differential equations just referred to, each equation being given both in its dimensionless and regular form, the dimensionless variables being indicated by placing a'bar over the symbol for the variable. It will be understood that each variable is thus represented by a pure number, indicated by the barred letter, multiplied by an appropriate combination of fundamental units which, for the pumping unit, are time, length and force, and which, for the electrical system, are time, voltage and charge, convenient units for these variables being selected as set forth in the first part of the following table, the remainder of the table indicating the notation to beused hereinafter.

o-Prime nzaover frequency or revolutions Q-Jlouimle t driving voltage on circuit per secon simulatim the en ine.

II.emth of se ment of rod, tube or oil. Unit of force-Force required to stretch a sucker rod segment by an amount P.

e, aver-Elastic modulus of thesucker rod,

oil and tube, respectively.

a, s.,, s ,-Cross sectional area of sucker rod, oil and tube, respectively. m-E quivaleut inertia of pump engine.

fFrictional coefficientof engine.

h-Positional constant for engine.

111,, we, a l-Weight per unit length of sucker rod, 011 and tube, respectively.

10,, wr weight .01 plunger and valve,

respectively.

It, k kfl-ViSCOU'S coeiiiciant'in expressions k('u+v), k (v+w), k211i giving viscous drag'per unit length between sucker rod and oil, between oil column and tube, and between tube and 011 surrounding tube. respectively.

q Upward pressure on bottom of tube due to oil between tube and easing -Circular frequency of driving volta e.

Maxirnum voltage applied to R line b con erator.

1'Charge required to raise voltage of a 1 condenser of capamtanceto a value c.

F Thus, w=E1-. .E.-, E0. i k-Reciprocal of capacity of the rod line, oil line and tubing line condensers,respectively.

W -W ,Elcctromotive force of batteries sunulatiug weight of plunger and valve, respectively.

R, R Fz-Hali theresistance in rod and oil lines, half the resistance in oil and tubing lines and resistance in tubing line, simulating viscous drag, in the expressions at the left.

(J llectromotivc force of-buttcry simulating upward pressure of oil on bottom of tube (0:110).

I,,- Coeflicient of self induction of choke simulating inertia of plunger.

L a -C0e1i"1cient oi self-induction of choke simulating inertia of valve end and tubing.

an, bf c,. =u-,., v, w", respectively, time derivatives of velocit or acceleration T-Time T- i' X,., Y Zg-Charge upon condensers simulating displacement oi rod, oil, and tube, respectively (X,.=+X,.; Y,.=1-Y,,;

' tubing lines simuating velocity of rod, oil and lube (U,.= 1U,.; V,.=p.1-V,.; W,.=urW,,). I

A. B,., Cn= T7,, V Tifitrespectivcly,

time derivatives of velocity, or ac- In the mathematical analysis of the pumping unit and electrical system which follows, the respective difierential equations are first derived by the method of finite differences and, from each such equation, the corresponding dimensionless equation is derived and follows the basic equation. In accordance with the method of finite difierences, the casing, tubing and sucker rod string are divided into a large number N of segments. During operation of the pumping unit, each segment of the sucker rod moves upward or downward in accordance with the mechanical laws governing the system.

In particular, considering the nth segment of the sucker rod, as defined by the dotted lines 32, 33, Figures 3, in an interval of time At, a volume of rod sunnt passes out at the bottom of the segment, and a volume of rod sun-nit passes into the top of the segment with the result that a volume s(un-un 1)At is removed from the segment. Accordingly, in this interval of time At, there is an increment Am in tensile stress in accordance with the following equation:

This equation has its counterpart in the electrical system of Figure 2 wherein electrical circuit unit 3:3 represents the 'nth segment of the pumping unit, circuit element 35 represents the top or the pumping unit, and circuit unit 36 represents the pump at the bottom of the well. The

segment 34 of the rod line includes an inductance R,,-Voltagesi mulatiug tens rod. The portion 3 5 of the rod line also includes v P,,Vo1tage i1; rod line simulating Q,,.Voltage simulating compression in I ion of tubing the voltage at station point 7L takes on the increment It will be noted that these two equations are of identical form and, upon comparing the dimensionless forms of these equations, the following relationships are obtained between the variables 101;, an and t of the pumping unit and the variables Pn, U12, and T of the electrical system, the symbol cr denoting the time ratio In the pumping unit 01. Figure 1, the downward velocity of the oil at the nth station is LL11, and the upward velocity of the oil is v with the result that there is an upward viscous drag upon the rod element of magnitude 76(Un-I-Un). From these data, the equation of motion of a rod segment 46 which straddles the segment between lines 32, 33 may be deduced, this segment extending distance above and below the nth station point. This segment has acting upon it a total tension pan on its bottom end, a total tension 11 on its top end,

the aforementioned upward viscous drag EHZLn-I-Dn and a downward weight wn. The resulting force produces a downward acceleration an at the up 1 er end of this segment in accordance with the equadiflerence Pn+I-Pn between the opposite ends accuse of segment 34 of the rod line causes a current U1 to flow. This current produces a voltage drop LrUn LrAn across the choke 37, the battery 39- produces an electromotive force Jr, and a voltage drop RUD is produced across resistance 40 which is coupled through transformer 42 with the oil line, this line, in turn, having a voltage drop of RVn across resistance 44. The action of this transformer produces a voltage drop across the parallel combination of the winding and resistance as which is equal to R(U1L+Vn). Accordingly, the equation for the voltage drop between one end and the other end of segment .34 of the rod line is given by the following equation:

It will be noted that Equations 6 and 7 are of similar form and, upon comparing the coefiicients and variables of the dimensionless forms of the equations, the following relationships are obtained between the coefficients and variables of the equations representing the electrical system and the equations representing the pumping unit of Figure 1.

J,=w,Z1- Eq. (8)

E LZE 9 R=kl 47p Eq. (10) v..="; v,. Eq. 11

a Eq. (12) Similar equations may be derived with respect to the compressive strain of the segments of the oil column and the motion of a general segment of the oil column. Thus, in a time interval At, a volume SoUnAt, Figure 4, of oil flows upwardly into the bottom of the nth segment, a volume So'Un1At flows out at the top of the nth segment and an increment SQ(Un-Un-1)Ai is accumulated within the segment. This increment is compressed and produces an increase new in the total oil pressure of an amount determined by the following equation.

(tn-n In segment 34 of the oil line, there is an inductance 53, a resistance 5! shunted by a winding 52 of a transformer 53, this transformer having a winding 54 which is connected in shunt with a resistance 55 in the tubing line, and a battery 56. The segment also includes a grounded condenser 5'! connected to the oil line, this condenser being shunted by a rectifier 58. It will be evident, by analogy to the reasoning employed in connection with the rod line, that a charge accumulates on condenser 51 in an interval of time AT as set forth in the following equation.

It will be noted that Equations 13 and 14 are of similar form and, upon comparing the coefiicients of the dimensionless forms of the equations, the following relationships are obtained between the coefficients .and variables of the equations representing the electrical circuit and the equations representing the pumping unit.

E.,=s.,e., Eq. (16) The equation of motion of the oil segment which extends a distance above and below the lower end of the nth oil segment, as is shown by Figure 4, is governed by the ri-1'1") +1 Eq. (17

In the electrical system of Figure 2, the voltage difference Qn-1---Qn between opposite ends of the segment 34 of the oil line causes a current V11. to flow through this segment of the oil line. This produces a voltage drop LOV :LoBn across inductance .53 and the battery 56 produces a voltage drop Jo. Furthermore, the aforesaid current produces a voltage drop RV across resistancep l i which, by the interaction of transformer 42 with the voltage drop RUn across resistance 40 produces a net voltage drop across the shunted transformer winding 43 and resistance 44 of R (Un-l-Vn) The current also produces a voltage drop across resistance 5| of Rlvn while resistance 55 has a voltage drop Rlwn produced across it by current flow through the tubing line. Accordingly, thevoltage drop across the shunted transformer winding 52 and resistance 51 is R1(V11.+Wn)- The total voltage drop between the ends of segment 34 of the oil line is, of course, the total of all the voltages produced as just described, that is 2 3: B.+ j U,.+v.)+ v.+ W.) +1

It will be noted that Equations 1'7 and 18 of similar form and, equating the coefiicients of these equations, the following relationships are attained between the coefficients and variables.

J.,=wal1- Eq. (19) Lo Eq. (20) It is known that the oil column cannot support a tensile stress. That is, although the oil can be and is compressed, it cannot be subjected to tension. Therefore, Equation 1'7 is subject to the restriction that q, the stress on the oil column, cannot become negative which would represent a tensile stress. This important limitation has not been taken into account prior to my invention, insofar as I am aware, and this limitation destroys the continuity of differential Equation 1'7 which practically makes impossible a mathematical solution of the equation. However, this limitation may be readily imposed upon the apparatus of Figure 2 and, to this end, I have connected rectifier 58 in shunt with the condenser 51. If the voltage Q is positive, the rectifier 58 is inoperative and the segment 34 of the oil line simulates the behavior of the oil column in the manner already described. However, the voltage at Q11, cannot become negative since the rectifier 53 will not permit point Qn to reach a potential which is less than ground potential. As a result, the segment 34 accurately portrays the actual behavior of the oil column in that the column can be subjected to compressive stress but not to tensile stress. This limitation is expressed mathematically by the following restrictions upon Equations 17 and 18.

immens are; 6.40 E (23) The movement of the tubing is governed generally the same equations as the movement of the sucker rod. Referring to Figure 5, and considering the 'nth segment of the tubing, as defined by dotted lines 60 and 6!, a volume of tubing Stw'nAt passes out of thebottom of the segment during an interval At, a volume of tubing shun-1M passes into the top of the segment and, accordingly, in this interval, there is an increment in tensile stress in accordance with the following equation.

charge upon condenser 62 is given by the follow;

WF'W'HMT Eq. (25

B g- 3 6."? I

of any typical segment of the tubing,

string, and oil column. The actual pumping unit may be divided into as many segments as desired,

The equation of motion of the nth tube segment which extends a distance above and below the segment just mentioned is determined by the total tension n+1 at the bottom of the segment, the tension m at the top of the segment, the viscous drag kll('vn+wn) produced by friction of the oil column against the tubing and an upward viscous drag kzlwn pro-' duced by oil or other fiuidgpositioned between the casing and the tubing. This latter term need only be considered for the particular segments wherein oil is included between the casing and tubing. Accordingly, the equation of motion of this segment is as follows:

t r Eq. (28) In the electrical system of Figure 2, the voltage drop R1|.Rn+1 between the ends of segment 34 of the tubing line causes a current W11 to flow through this segment of the line which produces voltage drops across inductance 63, battery 55, winding 54 connected in shunt with resistance '55, and fixed resistance ed as respectively indicated in the following equation:

R,.+1 n t n t+' l( n+ n) +R2W" The fixed resistance 64 is included only for those segments of the tubing wherein a stationary body of fluid is positioned between the casing and tubing. The following relationships between the pumping unit and electrical system follow from comparison of the dimensionless forms of Equa- In the foregoing description, it will be evident that the circuit unit 34 is an electrical analogue ticker rod each such segment being represented by a corre-' spondingcircuit unit 3 3 in the electrical system,

the accuracy becoming greater with an increasmg number of segments. In order to compensate for the fact that a finite difference analysis is accurate only up to an interval of one-half the element of extension, half-chokes not shown, may be inserted at the ends of the re- I spective rod, oil and tubing lines.

As stated, the circuit thus far described simulates the sucker rod-string together with its associated oil column and tubing. The'simulator also includes the circuit unit 35 representing conditions at thetop of the well and the circuitunit 36 representing conditions at the bottom of the well.

The top of the tubing string is anchored to the casing and to the ground so that it must remain stationary, that is, its displacement must be zero at all times. This condition is represented in the electrical circuit by leaving the tubing line unconnected at its top and so that no charge can accumulate. This condition is represented by the following equations:

zo=; Zo=0 Eq. (34) With respect to the oil line, the oil is discharged at the top of the tubing into a storage tank against a substantially constant head. That is, the pressure at the top of the oilline is constant. This condition is realized in the electrical system by inserting a battery it at the top of the oil'line to simulate this end condition. This condition is represented by the following equations:

The sucker rod string is driven at its top end by the drive mechanism which includes the prime mover, flywheel, crank and walking beam of Figure 1. Although I shall hereinafter disclose a simulator unit which takes into account the reaction of the sucker rod string upon the drive mechanism, which unit will be connected at the top of the rod line, for present purposes, the driving system may be assumed to be a positively acting mechanical system of one degree of freedom. The action of the drive mechanism upon the top of the sucker rod may, therefore, be represented by the following function, the terms of which represent, respectively, the resultant force applied to the top of the sucker rod, the lumped kinetic energy of the drive mechanism, the lumped potential energy of the drive mechanism, the dissipative function of the engine and the force or reaction which the sucker rod exerts upon the drive mechanism.

For present purposes, it is assumed that the reaction of the sucker rod system upon the drive mechanism is negligible, which is true in certain practical applications and where results are desired only to a predetermined degree of accuracy, so that the motion at the top end of the sucker rod string may be expressed as a sinusoidal function. As indicated, the reaction of the sucker rod system upon the drive mechanism is taken into account when the simulator for the drive mechanism to be hereinafter described is connected at the top end of the sucker rod line.

In the electrical system, an inductance II provides for the kinetic energy of the drive mechanism, a condenser 72 provides for the potential energy thereof, a fixed resistance 73 provides for the dissipative function and a generator 14 produces the sinusoidal drive voltage. Thus, the behavior of the circuit elements representing the drive mechanism is represented by the following equation:

Comparison between the Equations 36 and 37 yields the following relationships between the 12 parameters of the electrical system and pumping unit:

m Eq. (38) F f; q-

The end conditions for the bottom end of the pumping unit are determined by the nature of the pump used, its dimensions, and the material from which it is made. This downhole pumping unit is represented in the electrical system by the circuit unit 36. This unit includes, in the tubing line, batteries 15, E6 of opposite polarity connested in series with an inductance T! and a winding 18 of a transformer 19. A second winding of transformer 19 is connected to ground and through a rectifier ill to a winding 82 of a transformer 83, the junction between winding 82 and rectifier 8! being connected through a rectifier 8! to a conductor 84 of the oil line. A second winding 85 of transformer 83 has one terminal connected through a rectifier 86 to conductor 84 and to ground through a rectifier 81 of opposite polarity, the other terminal of winding 85 being connected, through an inductance 88 and a battery 89 to the lower end of the rod line.

The arrangement of the rectifiers is such as to provide the circuit of Figure 6 during the period representing the upstroke of the pump and the circuit of Figure 7 during the period representing the downstroke of the pump.

Referring now to Figure 1, during the upstroke, the working valve 2| in the plunger is closed and the plunger lifts the oil column upwardly at its.

own velocity so that v =u In Figure 6, the oil line is connected to the rod line through battery 89 and inductance 8-8 so that the following equation holds.

Furthermore, during the upstroke, the oil flowing into the working barrel l9, Figure 1, does not follow the plunger fast enough to exert appreciable pressure on its underside. The forces acting upon the plunger are, therefore, the weight of the plunger w the upward tension p in the rod and the downward pressure cm of the oil above the plunger. The. following equation, accordingly, describes the motion of the'plunger.

In Figure 6, the current flow through inductance 88 produces a voltage drop L AN and the battery 89 produces a voltage Wp, the sum. of these voltages being equal to the difierence in voltage (Q1vP1v) between the lower ends of the oil and rod lines. That is The following relationships follow from comparison of the dimensionless forms of Equations 42 and 43.

W =w rp Eq. (44) 2 L,,=2 L Eq. (45

wb ltq. (46) In the circuit of Figure 6, the voltage RN is equal to the voltage Wu of battery it minus the voltage Qb of battery 16 less the voltage drop LbCN across inductance TI. Thatis,

Comparison of the dimensionless forms of these equations yields the following relationships:

Wb=wb' P Eq. (48) L g Eq. (49

Thus, during the upstroke, the transformers i9, 83 and rectifiers ill, 86 and 8'1 of Figure 2 produce a circuit condition corresponding to Figure 6 which, in turn, electrically simulates the motion of and forces acting upon the plunger, traveling valve 2 I, and fixed valve 22 of Figure 1.

During the interval of time representing the downstroke, the transformer I9, 83 and rectifiers 8!, 86 and 81 of Figure 2 produce a circuit condition corresponding to Figure 7. The downward velocities of plunger l9 and the end of tube 13 being u and w respectively, a volume of oil (Sr-I-So) (u -w At is displaced upwardly through valve 2i during an interval of time At. This upwardly displaced oil occupies an equal volume so(v +u )At in the column above the plunger. Accordingly,

In the circuit of Figure '7, the transformer 33 is wound with 2, turns ratio so that, when the voltage and current in winding 85 are and UN, respectively, the voltage and current in winding 82 are Q11. and

a u e and W1, respectively. The nodal equation for the junction Qn is therefore,

If the dimensionless forms of Equations 51 and 52 are derived and their coefiicients compared, certain of the same relationships already derived will be obtained between the coefficients and variables of the two sets of equations. It will be further noted that the two equations are of similar form so that the circuit of Figure 7 accurately simulates the aforesaid end condition during the downstroke period.

Since oil passes freely through the valve 2| during the downstroke, the pressure intensity in the oil below and above plunger [9 is nearly equal. The cross-sectional area of oil in the barrel, however,' exceeds that above the plunger in the ratio and, consequently, the upward force on the lower end of the plunger exceeds that on the upper end of the plunger by Further, the plunger is subjected to the downward force of its weight w and the tension p in the sucker rod string; The equation of motion of the plunger during the downstroke is, therefore,

pq1v "PN f p N/9 q- Similarly, in Figure 7, the branch equation for the line extending from Pn to ground is as follows, upon consideration of the turns ratio of the transformers set forth in the discussion of Equation 51.

W,,-- ,'-P,=L,U, Eq. (5;

It will be noted that Equations 53 and 54 are of the same form so that the electrical simulator accordingly portrays the movement of the plunger during the downstroke. If the dimensionless forms of these equations are derived and their coefficients are compared, certain of the relationships already obtained between the coemcients and variables of the two systems will be found to hold.

Further during the downstroke period, the forces acting upon stationary valve 22 are the weight of the valve we, the downward pressure q of the oil in the barrel of the plunger, an upward pressure qb on the bottom of the valve head,

and the tension TN in the metal at the end of the tube. This valve, accordingly, has the following equation of motion.

In Figure 7, the branch equation for the line extending from point R11 to ground is as follows in consideration of the turns ratio of transformers 19 and 83.

It will be noted that Equations 55 and 56 are of the same form and, hence, the circuit accurately simulates the operation of valve 22 during the downstroke period. If the dimensionless forms of these equations are derived and their coeflicients compared, certain of their relationships already derived will be found. I

gear-sea In the operation of the circuit, the circuit of Figure 2 is used to simulate conditions existing in an actual pumping installation. In this actual installation, most of the constants are known and these constants are utilized by substituting them in the equations previously derived to deter mine the corresponding parameters of similar parts of the electrical analogues of theseconstants. Before making these substitutions, the values of o, and E are assumed from which data and p may be readily calculated. In this connection it will be noted and that the cross sectional areas of various parts of the actual pumping system as well as their elastic modulus are known from which it follows that p is readily calculated once a value for E is assumed. It is further necessary to assume a value for the segment length Z which may conveniently be the length of one section of the sucker rod string and, in the particular circuit shown by Figure 2, the equivalent driving voltage Q, and the maximum voltage a are assumed quantities from which data the value of 1- may be readily calculated from the equation =E'r. When the simulator hereinafter to be described is utilized to produce the driving voltage, the quantities o, 11/, and 1- are obtained by measurements of the simulator output. Once these constants have been assumed and calculated, as stated, the

known constants of the actual pumping system are substituted in the equations to provide corresponding parameters for the constants of the electrical system. As an example of such substitution, referring to Equation 8, the weight wt of a unit length of sucker rod is known when the actual pumping system is constructed. The quantities Z, T, p, are assumed or calculated as just indicated and, upon substitution of these values in Equation 8, the quantity J 1', that is, the electromotive force of a battery 39 in the rod segment is readily calculated. Similarly, with respect to Equation 9, since the weight of a rod segment wr is known and the quantities Z, a, p, and g are known or calculated, the value L! of an inductance 31 in the rod line is readily calculated from Equation 9. In an entirely similar fashion, the parameters of the electrical system set forth in Equations l0, l6, 19, 20, 21, 26, 2'7, 30, 31, 33, 34, 35, 38, 39, 40, 44, 45, 48, i9 and 50 are readily calculated since the righthand side of each of these equations contains only the known or assumed quantities Z, c, 9, 11-, 1-, a, p and g which are assumed or calculated in the manner previously set forth together with a constant which is readily measurable or known on the actual pumping system. This enables the entire circuit of Figure 2 to be constructed, each component having a value as determined by the respective foregoing equations.

When this is done, the relationships set forth in Equations 3, 4, 5, 11, 12, 15. 22- and 32 hold between the variables of the electrical system and the variables of the actual pumping system. The values of the assumed quantities are then changed until the electrical system behaves in a manner analogous to the actual pumping system. Thereupon, the effect of any changes desired in the parameters of the mechanical system may be accurately predicted by noting the effect of corresponding changes on analogous electrical system. This enables operating conditions to be determined so that the dimensions and physical condii6 tions of the actual pumping system are at an optimum to provide most efficient operation with resultant elimination of difficulties due to sucker rod breakage and malfunctioning of other parts of the apparatus.

Where all the necessary constants are not known in the actual system so that the corresponding electrical parameters of the electrical system are correspondingly unknown, the known values are used in the manner previously set forth to determine the constants of the corresponding parts of the electrical circuit. Reasonable values are assumed for the other known constants and corresponding values are set in the simulator by use of the aforementioned equation.

The simulator and its operations are then studied and conclusions are drawn as to the extent to which the performance depends on the unknown constants for which values had been assumed. Usually it will be found that some of these are critical. Others may lie anywhere within a fairly wide range without appreciable effect on the performance of the simulator. Also, operation of the simulator suggests measurements to be taken on the prototype which either will permit computation of those critical constants or will serve as indices to the performance of the prototype.

In Figure 2, the transformers 42 and 53 of each segment were assumed to be ideal transformers having a perfect one-to-one coupling ratio. Where this assumption leads to any appreciable error, the transformers i2 and 53 may each be replaced by the unit shown in Figure 8 which provides, in effect, an ideal transformer. In this figure, the output of an operational amplifier 91 is applied across resistance 48 or" a segment in the rod line and the output of an operational ampliher 32 is applied across resistance 5-4 in the oil line. The input of amplifier 32 is coupled to resistance ii through an impedance 93 and ampliher 92 is provided with a feedback impedance 84. Similarly, the input of amplifier 9] is coupled to resistance i l through an impedance and this amplifier is provided with a feedback impedance 98. Each amplifier produces a powered output which reflects its input potential to the resistance as or 44 in the line to which the amplifier output is connected. However, the amplifier input does not draw appreciable current from the line to which its input circuit is connected. As a result, the circuit of Figure 8 is the full equivalent of transformer d2 of Figure 2 and has the advantage that no current is drawn from the lines by either of the input circuits.

I shall now describe a simulator unit for the drive mechanism N3 of Figure 1. When this circuit is used, it replaces the inductance H, condenser E2, resistance 23 and generator M of Figure 2, the simulator producing a voltage output which is representative of the driving force applied to the top of the sucker rod strin by the mechanism it and also taking into account the reaction of the downhole pumping system upon the drive mechanism.

In this simulator unit, an electrical network is provided which satisfies the differential equations and positional equations of the various parts of the drive mechanism H3. The notation utilized hereinafter has no relation to that previously used and is to be considered independently of the preceding description. For convenience, the following equations are referred to Figure 9, which is a schematic representation of the drive mechanism ll] of Figure 1. The motor armature is 17 designated as rotor (1) the flywheel is designated as rotor (2) and the crank 26 is designated as rotor (3), the subscripts in the following notation system designating thejrotor referred to. The following notation is utilized in th equations:

torque J (51 and a friction torque F101. Hence,

of rotary Similarly, the rotary equations for the flywheel and crank are The equation of motion of the Walking-beam is (1T3 cos a cos afar. sin a sin 0,1 P=J32);+F.4

The following equations express obvious geometrical relations between positions of crank, pitman, and walking-beam Z cos Clo-Z cos a=a sin 6 sin Eq.(62)

Z sin a-sin a =r +ar cos 0 -11 cos 6 Due to stretching of the chain by which the motor drives the flywheel, its length increases by and this requires a tension T1 given by T =K (1' 0 R 0 Eq. (64) Where K1 is an elastic constant. Similarly T =K (r 0 R 0 Eq. (65) The last two equations represent the energy storage or spring action of the engine, similar effects for the pitrnan and elastic deflection of the walking-beam being considered negligible although these latter can be readily taken into account, if

desirable. s 7

Inspection of these nine equations shows that there are nine variables involved, namely,0 0,, 0,, 0,, a, N, T T T where M and P are not included as unknowns because they represent external actions on the system and must, so far as the simulator is concerned, be given. Actually, P is supplied by the simulator for th pumping string for producing an output in accordance with each to zero potential.

and M must be applied by external means. Ac-

cordingly, there are sufiicient equations to determine all the variables noted.

In accordance with the invention, each of the foregoing equations is rearranged so that a single variable appears at the righthand side of each equation it being immaterial how many variables appear at the lefthand side of each equation. These new equations are not mathematical solutions of the first equations since more than one variable appears in the lefthand side of the equations. These rearranged equations are as follows:

1 arc sin U cos a l cos a+r sin =0 In constructing the simulator, 1 provide'a unit of the A group of equations. Each variable is fed to as many units as required as an output without disturbing the linearity of the circuit from which the output is taken. Theoutput P represents the force applied to the polished rod 23 taking into account the reaction of the downhole apparatus shown in the drive mechanism. In the following discussion, therefore, it will be assumed that outputs are supplied as needed to the respective equation units and these outputs will be actually supplied by the coupling units to be described after the individual equation units are set forth. Figure 11 illustrates the manner in which the various equation units are connected together by the coupling units.

input circuit by a unit resistance W2 and input terminals ml being connected to the input circuit by a resistance I03 of ohmic value The amplifier has a feedback resistance we 01 ohmic value which is shunted by a condenser we of capacitance Jr. The action of the amplifier is such as to drive the voltage at point me substantially Accordingly, a current This totalit current M T1'r1 through divides and flows through resistance Iii l shunted by condenser I05, the respective currents through these components being F 01 J 2340 where p represents the operator Accordingly, the expression M -T1'r1, is equal to the expression cl ps-F which is equivalent to Equation 57A.

The same circuit is utilized to provide the variable w of unit 2 except that the circuit com ponents have the values indicated in Figure 13 rather than the values indicated in Figure 12. By comparison of the two circuits it will be readily apparent that the output 0 of unit 2 is equivalent to Equation 58A. In similar fashion, the circuit of Figure 12 is used in unit 3, Figure 14, to provide the output co the circuit components having the values shown in Figure 14 rather than the values shown in Figure 12 or 18. From a comparison of the circuits, it will be evident that the output 01 of the circuit of Figure 14 satisfies Equation 59A.

The circuit of unit 4 is shown in Figure 15. This circuit includes a high gain amplifier I08 having its input terminals connected in shunt with a fixed resistor Its, one terminal of which is connected to one of a. set of input terminals I III, the other terminal of this set being grounded. The other terminal of resistance It?! is connected through a resistance I I I to the contact of a potentiometer H2 having a grounded center tap H3. One fixed terminal of potentiometer H2 is connected to one of a set iI-I of input terminals, the other terminal of the set being grounded and said fixed terminal is also connected through a unit resistance IIS to the input circuit or a high gain amplifier IE8 having a unit feedback resistance I IT. The output of the amplifier I I6 is fed to the other fixed terminal of the potentiometer. Aset of output terminals H3 are connected to the contactor of a potentiometer III! which has a grounded center tap I28; and which is shunted by a battery, not shown, having an electromotive force of two units. The output of amplifier I08 drives a servomotor I22 which is mechanically connected to the contactors of both potentiometers H2 and II9.

In operation, assuming the indicated inputs are applied to inputs I I0 and I I 3, amplifier I08 drives motor I22 until the contactor of potentiometer II2 reaches a point P where its voltage is equal to the voltage at point S. When this occurs, the motor stops and the following equation follows from the indicated relationships between the potentials at the potentiometer contactor and terminals.

At the same time, contact Q is also positioned at the distance a: from the midpoint of potentiometer HQ with the result that the potential across terminals H8 is When the value of is substituted in this expression from Equation 66, the Equation 60A is satisfied. The function or the amplifier H6 and unit resistances H5, H1, is

to change the sign of the voltage applied to its input terminals so that either negative or positive potentials may be obtained from the potentiometer depending upon the sign of the voltage impressed upon terminals III). Similarly, since potentiometer I I9 has its midpoint grounded permits either positive or negative output voltages to be withdrawn from output terminals i I8.

The circuit for producing the output N of unit 5 is similar to that of Figure 15 and is shown in Figure 16, it being understood that the circuit components having the values shown in Figure 16 rather than the values shown in Figure 15. Upon comparing the operation of the two circuits, it will be evident that the output N of Equation 61A is produced by the circuit of Figure .16 when the indicated potentials are applied to input terminals IIII and N 1.

The output 6, of Equation 62A is produced by unit 6 shown in Figure 1'7. In this circuit, three sets of input terminals I2 i, I25 and I26 are connected, respectively, through unit resistances I27, I28 and I29 to the input circuit or" a high gain amplifier I36 having a unit feedback resistance I3I. Amplifier I39 adds the indicated inputs appearing at inputs I24, I25 and I26 and applies the sum of these inputs to a high gain amplifier I32 through a unit resistance I33. The junction between the amplifier input and resistance I33 is connected through a unit resistance I34 to the contactor of a sine potentiometer I35 having the indicated voltages applied to its fixed terminals I36 and I 31, the contactor being driven by a servomotor I38 which receives the output of amplifier I32. Accordingly, the motor I38 moves the contactor of potentiometer I35 to a position P where the sine of angle 6, is equal to the sum of the inputs fed to amplifier I32 through resistance I33. Thus, the angle 0, satisfying Equation 62A appears as a shaft rotation of the contactor of potentiometer I35. The motor I38 also drives the contactor of a cosine potentiometer I40 to a position Q representing the angle 0,, this potentiometer having the indicated voltages supplied to its fixed terminals HI and I52. As a result, the voltage a cos. 0 appears across output terminals I43.

The same circuit is utilized in unit 7, Figure 18, to produce the output a of Equation 63A, it being understood that the circuit components have the values indicated by Figure 18 rather than the values indicated by Figure 1'7. When the indicated voltages are applied to terminals I24, I25 and I26, the output a. is produced as a shaft rotation of the contactor of potentiometer I35 and the output Z cos it appears across output terminals I43.

The output T1 of Equation 64A is produced by the circuit of Figure 19, unit 8. In this circuit,

a set of input terminals I45 is. connected in series with a variable resistance I46 and a rate servomotor I so that the motor can be adjusted till its speed is T w Wh8I1 the input fo is applied to terminals I45. A second set of input terminals I48 is connected in series with a variable resistance I 49 and a rate servomot'or I50 so that the speed of this motor can be adjusted to represent R w when a voltage m is applied to terminals I48. As a result, the shaft rotation of motor I41 is 130, and the rotation of the shaft of motor I50 is R 0 These shaft rotations are added by a differential I5I which drives the contactor of a potentiometer I52 having a grounded center tap, a voltage K1 being applied to a fixed terminal, I53 of this potentiometer .anda voltage.

21 K1 being applied to the other fixed terminal I54 of the potentiometer. As a result, a voltage T1 satisfying Equation 64A appears at output terminals I55 which are fed by the contactor of the potentiometer. The circuit of Figure 20, unit 9, is

similar to that of Figure 19 except that the circuit components have the values indicated in Figure 29 rather than those indicated by Figure 19. Accordingly, when the indicated inputs are fed to terminals I55, I48, an output Tz satisfying Equation 65A appears at the terminals I55.

In this fashion, the outputs of all the equation units are provided assuming that each unit re-v ceives the inputs specified in Figures 12 to 20, inclusive. The manner in which the various outputs are supplied to the proper units is shown in the block diagram, Figure 11, and these units are arranged as shown in Figure 11 together with the coupling circuits now to be described.

The inputs required for unit 1 are M and Tl. The potential M is supplied by a pulse generator I60, the wave form of the pulse produced by the generator representing the driving torque on the motor as a function of time, which may be readily accomplished in the case of an electrical motor. Where a gasoline or diesel engine is used, the pulse generator produces a wave representative of the indicator diagram of the engine translated into terms of torque. The voltage from the pulse generator is supplied to unit 1. Unit 1 also requires, as an input, the quantity T1. This input is produced by transforming the output T1 a pearing at the terminals I55 of unit'B, Figure 19, by the coupling unit 8 1 of Figure 21. This unit includes a high gain amplifier I6! having an odd number of stages to which the input T1 is applied from input terminals I62 through a unit resistance I63, the amplifier being shunted by a unit feedback resistance I64; As a result, the quantity Ti appears at output terminals I55 as a powered output and is fed to the proper terminals of unit 1. Thus, the amplifier produces all the power needed to supply unit 1 but no appreciable input current is drawn by this amplifier to disturb the linearity of the circuits of unit 8.

The inputs required for unit 2 are T1 and T2 The input -T1 is supplied to terminals I from terminals !55 of coupling unit 81 1, Figure 21. A similar coupling unit 9 2, Figure 11, is of identical structure with the circuit of Figure 21 and this unit transforms the outputs appearing at terminals I55,Figure 20, into the input T2 required at the terminals IM, unit 2, Figure 13.

The inputs required for unit 3 are T2 and N. The potential T2 is provided by coupling unit 9+2 which has just been described and the input N for terminals WI of unit 3, Figure 14, is supplied by a coupling unit 5 3, Figure 11, of identical construction to coupling unit 8-)1, Figure 21, this unit transforming the output N appearing at terminals H8, unit 5, Figure 16, to the required input N.

The inputs required by unit 4, Figure 15 are (J,p+F,)w,bP and a cos.(0,+a). The first input is provided by coupling unit 6+4, Figure 22. The output 9, of unit 6, Figure 17, appears as a shaft rotation. The rotating shaft is coupled to the contactor of a potentiometer 165 supplied with a unit voltage across its fixed terminals, thus producing a potential 0, at the contactor of the potentiometer. The contactor is coupled to a high gain amplifier I61 through a unit capacitance 168, the amplifier being provided With a unit feedback resistance I69. ,Accordingly.. the.

and a condenser I1! of ohmic value J4. Thus, a conductor I12 connected to the input circuits of a high gain amplifier I13 receives, from this network, a current J, w, from condenser HI and a current ,w, from resistance I10. A terminal I 15 is connected to the top of the rod line of the simulator unit of Figure 2 and thus receives a voltage P representing the action of the downhole pumping unit. The voltage applied from coupling unit 6 4 to terminal I15, in turn, s representative of the driving force applied to the top of the sucker rod string by the drive mechanism simulator unit. Thus, at point P, the driving effect of the simulator is transmitted to the sucker rod simulator and the reaction of the sucker rod simulator is transmitted to the driving mechanism simulator. Terminal I15 is connected to conductor I12 by a fixed resistance I15 of ohmic value so that; when the voltage at conductor 112 is driven substantially to zero by the action of unit feedback resistance I11, the conductor I12 receives a voltage I)? from resistor I13. The current flowing through condenser 11! and fixed resistances I15, I16 are added by amplifier I13, the output of which produces the indicated voltage across terminals I18 for application to the input terminals I II] of unit 4, Figure 15.

The input a cos.(0,-|-a)for the input terminals I I 4 of unit 4 is produced by coupling unit 6, 7+4, Figure 23. This unit'includes a mechanical differential I19 which produces a shaft rotation equal to the sum of the shaft rotation 11. appearing as an output of unit '7, Figure 16, and the shaft rotation 0, appearing as an output of unit 6, Figure 17. The mechanical differential is c0nnected to the contactor of a cosine potentiometer I having voltages a and a supplied to the respective fixed terminals thereof. As a result, an output voltage a cos.(0,+a) is generated which, through the medium of a sign changing unit including a high gain amplifier IBI, a unit input resistance 182, and a unit feedback resistance I83, is converted into the required input voltage appearing at output terminals EM which are connected to the terminals I14 of unit 4, Figure 15.

The inputs required for unit 5, Figure 16, are T, and cos.(0,a). A coupling unit 4+5, Figure 11, of identical construction to coupling unit 8 1, Figure 21, transforms a voltage -T, appearing at output terminals lid of unit a, Figure 18, to the voltage T, required at the input terminals Hit of unit 5, Figure 16. The input for terminals I It is produced by a coupling unit 3, '1 5, Figure 24, including a mechanical differential 85 having the shaft rotation a from the output of unit '1, Figure 16, applied to one side thereof. The coupling unit also has a set of input terminals H85 which receive an input -w from the terminals $9 of unit 3, Figure 14. The voltage fed to terminals is passed through a variable resistance 1581 to a rate servomotor which is adjusted to rotate at a speed 6, by

proper adjustment of Variable resistance I51, this servomctcr, in efiect, transforming the electrical quantity representing velocity into a mechanical shaft rotation representing displacement. The output of diderential its is fed to a continuous cosine potentiometer E39 which produces an output cos.(0,,a) which, after passing through a sign changing unit comprising a high gain amplifier let, a unit input resistance llll and a unit feedback resistance 1S2, appears as the proper output at terminals to be fed to the terminals 5 it or unit 5, Figure 16.

The inputs required for unit 6, Figure 17, are indicated at terminals 24, i25- and 125 on this A sign changer of identical construction to unit 8- 1, Figure 21, and designated 7-)6 in Figure ll converts the voltage -l COSJZ appearing at terminals M3 of unit 7, Figure 16, to the required voltage to be fed to terminals like of unit 6, Figure l'l'. The voltage Z 395. is a constant voltage and may be supplied directly to terminals its by any suitable source. Preferably, the negative or" this voltage is produced across a potentiometer and fed through a sign changer to the terminals iz i of Figure 17 so that ample power is available to supply the circuit of unit 8, Figure 17. The voltage -r,, coax) is supplied by coupling unit 3- 6, Figure 25, wherein the shaft rotation 0, produced by rate servomotor Hit of coupling unit 3, 7 5, Figure 24, is applied to the contactor of a continuous cosine potentiometer its, the fixed terminals of which are supplied with voltages r, and 2",, respectively. This produces a voltage r3 005.03 at the oontactor of the potentiometer which, through the medium of a sign changer including an amplifier I85, a unit input resistance lab, and a unit feedback resistance let, produces the required output voltage at terminals its for application to the terminals I25 of unit 6, Figure 17.

in regard to the inputs. required for unit '7, Figure 13, the required input for ternnnals E25, that is, 1", cost is available at the terminals let of coupling unit 3' 6-, Figure 25. The input a cost, for terminals i125 taken from terminals its of unit Figure l'l', a sign change or coupling unit fi 7 of identical construction with the coupling unit 8 1., Figure 21, being provided to change the sign. of the voltage in the required manner. The input required for terminals I 24- is a constant and may either be supplied directly to the terminals or, alternatively, the negative of the required voltage may be produced at the contactor of a potentiometer and this voltage fed through a sign changer unit to provide adequate power for operating the circuits of unit 7., Figure 16. The input -w, for terminals M5 oi unit 8, Figure 17, is supplied directly from the terminals 99 of unit 1, Figure 12.

The input w, for the terminals 143 or unit 8, Figure 19, is supplied directly from the terminals at of unit 2, Figure 13, adequate power'being available at both sets of output terminals to supply the circuits of units 8 and 9. The input w, for terminals MS of unit 9, Figure 20, is supplied directly from the terminals 89 of unit 3, Figure 14. Finally, the input for terminals MB of unit 9, Figure 29, is supplied directly from terminals 28 of, unit 2, Figure 13, adequate power being available at both sets of output terminals to'supply the circuits of Figure 20'.

In this manner, the inputs are supplied to each equation unit by the described coupling circuits with adequate power to operate the units thus supplied. Furthermore, no current is drawn from the equation units which supply outputs to other 24 units through the coupling devicm. The voltages and currents in the system of Figure 11 satisfy the differential equations of motion (Equations 57 to of the drive mechanism for the pumping unit and, hence, the voltage impressed at the top of the rod line of Figure 2 reflects the efiect of the drive mechanism for the pumping system. Furthermore, the voltages appearing at the top of the rod line in the operation of the sucker rod simulator are reflected into the cir cuits of the drive mechanism simulator so that the reaction of the pumping system shown in drive mechanism is accurately taken into account. When connected together in the manner already described, the two systems provide appa ratus which is an electrical analogue of the entire pumping system including the drive mechanism, the downhole pumping unit, the rod and tubing strings, and the oil columns both between the sucker rod and tubing and between the casing and tubing.

While the invention has been described in connection with present, preferred embodiments thereof, it is to be understood that this description is illustrative only and is not intended to limit the invention, the scope of which is defined by the appended claims- I claim:

1.. In apparatus for electrically simulating a segment of. a downhole pumping unit including a tubing segment, a rod. segment, and an oil column segment, in combination, an electrical line for each of said segments, each line including a condenser representing displacement of the segment, an inductance representing the inertia of the segment, and a Voltage source representing the weight of the segment, a resistance in the rod line, a resistance in the tubing line, a pair of resisters in the oil line, a coupling circuit interconnecting one of said pair of resistors and the tubing line resistance to simulate viscous drag between the tubing segment and the oil column segment, a coupling circuit interconnecting the other of said pair of resistors and the red line resistance to simulate viscous drag. between the rod segment and the oil column segment, and a rectifier connecting the oil line to a point maintained at a reference potential to prevent the oil line voltage from falling below said reference potential.

2. In apparatus for electrically simulating a segment of a downhole pumping unit including a tubing segment, a rod segment, and an oil column segment, in combination, an electrical line for each of said segments, each line including a condenser representing displacement of the segment, an inductance representing the inertia of the segment, and a voltage source representing the weight of the segment, a resistance inthe rod line, a resistance in the tubing line, a pair of resistors in the oil line, a coupling circuit interconnecting one of said pair of resistors and the tubing line resistance to simulate viscous drag between the tubing segment and the oil column segment, a coupling circuit interconnecting the other of said pair of resistors and the rod line resistance to simulate viscous drag between the rod segment and the oil column segment, and a second resistance in the tubing line representative of the friction. of a stationary body of oil outside the tubing segment.

3. In apparatus for electrically simulating a downhole pumping system including a sucker rod. string, tubing: surrounding said sucker rod string, an oil column between" said rod and said tube, and a pump at the bottom oi said system,

incombination; a plurality of units, one for each segment of the pumping system, each unit including an electrical line simulating each of a tubin segment, a, rod segment, and an oil column segment, respectively, each of said lines including a condenser representing displacement of the segment, an inductance representing the in ertia of the segment, and a voltage source representing the weight of the segment each of said units further including a resistance in the rod line and a resistance in the tubing line, a pair of resistors in the oil line, a coupling circuit in terconneoting one of said pair of resistors and the tubing line resistance to simulate viscous drag between the tubing segment and the oil column segment, a coupling circuit interconnecting the other of said pair of resistors and the rod line resistance to simulate viscous drag between the rod segment and the oil column segment, and a rectifier connecting each segment of the oil line to a point maintained at a reference potential to prevent the oil line voltage from falling below said reference potential; means connecting corresponding lines in each of said units in series relation to simulate operation of said downhole pumping system: and means connected to the line simulatingthe uppermost rod segment'for simulating the driving mechanism at the top of the pumping system; and means interconnecting the lines simulating the lowermost rod, tubing and oil column segments for simulating the operation of the pump at the bottom of said system.

4. Apparatus for producing the effect of an ideal transformer which comprises, in combination, a pair of electrical lines each including an impedance, means for causing a current to flow through each of said impedances, a high gain amplifier having its input circuit connected through a coupling impedance across one of said line impedances and its output applied across the other line impedance, a second high gain amplifier having its input-circuit connected through'a second coupling impedance across the other line impedance and its output applied across said first line impedance, and a feedback resistance connected in circuit with each of said high gain amplifiers, each of said coupling impedances and said feedback resistors being of equal ohmic value.

5. A simulator for a pumping unit driving mechanism including a prime mover, a flywheel, a crank, a walking bearndriven through a pitman by the crank, and means connecting the prime mover to the flywheel and the flywheel to the crank, which comprises, in combination, a

plurality of means for combining mechanical and electrical quantities as dictated by the difierential equations of motion and spatial relationships of the components of the drive mechanism so as to produce outputs representative of the system variables, means for'applying a driving voltage representing prime mover characteristics to one or said plurality of means, means for feeding said outputs as poweredv inputs, to said plurrlity of means where such outputs are required, and means for applying a voltage representing the reaction of a downhole pumping unit to one of said plurality of means, the resulting voltage at said point of application of said last-named voltage simulating the driving force of said mechanism.

6. A simulator for a pumping unit driving mechanism including a prime mover, a flywheel, a crank, a walking beam driven through a pitman by the crank, and means connecting the prime mover to the flywheel and the flywheel to the crank, which comprises, in combination, a plurality of means for combining mechanical and electrical quantities as dictated by the differential equations of motion and spatial relationships of the components of the drive mechanism so as to produce outputs representative of the angular velocity of the prime mover, flywheel, and crank, the tension in said connecting means and said pitman, the torque produced by the crank and the angular position of the walking beam and pitman, means for applying a driving voltage representing prime mover characteristics to one of said plurality of means, means for feeding said outputs as powered inputs to said plurality of means where such outputs are required, and means for applying a voltage representing the reaction of a downhole pumping unit to one of said plurality of means, the resulting voltage at said point of application of said last-mentioned voltage simulating the driving force of said mechanism.

7. A simulator for a pumping system comprising driving mechanism which includes a prime mover, a walking beam, and means connecting said prime mover in driving relationship to said walking beam, said pumping unit including a sucker rod string driven by the walking beam, tubing surrounding said sucker rod string, and a column of oil between said tubing and said sucker rod string, incombination, a plurality of means for combining mechanical and electrical quantities as dictated by the differential equations of motion and spatial relationships of components of the drive mechanism so as to produce outputs representative of the system variables, means for applying a driving voltage representing prime moving characteristics to one of said units, means for feeding said outputs as powered inputs to said plurality of merns where such outputs are required, a first electrical circuit whose parameters are representative of the variables and the constants of the downhole pumping unit, a second electrical circuit for simulating the operation of a pump mounted at the bottom of said tubing, means for s mulating connection of said tubing to ground at the surface of the well, means for simulating the constant pressure at which oil is discharged from the top of the well, and means for applying the output of one of said plurality of means to said first electrical circuit so that saidfirst circuit is driven by a voltage corresponding to the action of the drive mechanism, said first circuit, in turn, producing an electrical reaction upon the plurality of means of the driving mechanism.

8. A simulator for a pumping unit driving mechanism including a prime mover, a flywheel. a crank, a walking beam driven through a pitrnan by the crank, and means connecting the prime mover to the flywheel and the flywheel to the crank, which comprises, in combination; a plurality of means for combining mechanical and electrical quantities as dictated by the diiierential equations of motion and spatial relationships of the components of the drive mechanism so as to produce outputs representative of the angular velocity of the prime mover, flywheel, and crank, the tension in said connectin means and said pitman, the torque produced by the crank, and

assists 2-? where such outputs are required; apparatus for electrically simulating a downhole pumping system including a sucker rod string, tubing surrounding said sucker rod string, and an oil column between said rod and said tube, said apparatus comprising, in combination, a plurality of simulator units, one for each segment of the pumping system, each unit including an electrical line for each of a tubing segment, a rod segment, and an oil column segment, each line including a condenser representing displacement of the segment, an inductance representing the inertia of the segment, and a voltage source representing the weight of the segment, a resistance in the rod lin and the tubing line, apair of resistors in the oil line, a coupling circuit interconnecting one of said pair of resistors and the tubing line resistance to simulate viscous drag between the tubing segment and the oil column segment, a coupling circuit interconnecting the other of said pair of resistors and the rod line resistance to simulate viscous dra between the rod segment and the oil column segment, and a rectifier connecting the oil line to a point maintained at a reference potential to prevent the oil line voltage from falling below said reference potential; a unit for simulating the end conditions at the top of the pumping system; a unit for simulating the operation of the pump at the oottom of. said system; and a coupling circuit connecting one of said circuit units to the line representing the top segment of the sucker rod.

9. The combination in accordance with claim 1 wherein said first-mentioned coupling circuit comprising a transformer having one winding thereof connected across the resistance in the tubing line and the second winding thereof connected across one of said pair of resistors, and wherein said second-mentioned coupling circuit comprises a transformer having one winding thereof connected across the resistance in the rod line and the second winding thereof connected across said other resistor.

10. In apparatus for electrically simulating a downhole pumping unit including, a tubing segment, a rod segment, and an oil column segment, in combination, an electrical line for each of said segments, each line including a condenser repre senting displacement of the segment, an inductance representing the inertia of the segment, and a voltage source representing the weight of the segment, a resistance in the rod line, a resistance in the tubing line, a pair of resistors in the oil line, a coupling circuit interconnecting one of said pair of resistors and the tubing line resistance to simulate viscous drag between the tubing segment and the oil column segment, a coupling circuit interconnecting the other of said pair of resistors and the rod line resistance to simulate viscous drag between the rod segment and the oil column seg ent, and a voltage limiting circuit to maintain the magnitude of the voltage on the oil line. within preselected limits whereby the voltage on said oil line is prevented from assuming a value outside, of said preselected limits representative of tension in the oil column.

11. The combination in accordance with claim 3. wherein said means connected to the rod line simulating the uppermost rod segment. comprises a generator connected to the rod line for producing a driving voltage representative of the force of the driving mechanism at the top of the pumping system, and further comprising a voltage source connected to the line simulatin the uppermost oil column segment to simulate the pressure at which oil is discharged from the tubing.

12. The combination in accordance with claim 3 wherein said pump includes a plunger, a traveling valve and a stationary valve; wherein said last-mentioned means comprises a voltage source simulating the weight of said stationary valve, an inductance simulating the inertia of said valve, and a primary winding of a first transformer connected in series relation with the line simulating the lowermost rod segment; a voltage source simulating the weight of said traveling valve and plunger, an inductance simulatin the inertia of said traveling valve and plunger, and aprimary winding of a second transformer connected in series relation with the line simulating the lowermost tubing segment; a voltage source connected in series relation with the parallel connected secondary windings of said first and second transformers and with the line simulating the lowermost oil column segment whereby the current supplied to the secondary windings of said transformers simulates the velocity of an oil column above said traveling valve, said lastmentioned voltage source simulatin the stress of. an oil column above said traveling valve, the turns ratio of the primary and secondary windings of said first transformer being the ratio between the cross-sectional area of said rod and the cross-sectional area of said oil column, the turns ratio of the primary and secondary windings of said second transformer being the ratio between the sum of the cross-sectional areas of said sucker rod and oil column and the crossscctional area of said oil column; and means for preventing current flow through said transformers during periods representing the upstroke of said pump.

13. Apparatus for simulating a downhole pump including a plunger, a traveling valve and a stationary valve comprising, in combination; a voltage source simulating the weight of said stationary valve, an inductance simulating the inertia of. said valve, and a primary winding of a first transformer connected in series relation; a voltage source simulating the weight of said traveling valve and plunger, an inductance simulating the inertia of said traveling valve and plunger, and a primary winding of a second transformer connected in series relation; a voltage source connected in circuit with the parallel-connected secondary windings of said first and second transformers whereby the current supplied to the secondary windings of said transformers simulates the velocity of an oil column above said traveling valve, said last-mentioned voltage source simulating the stress of an oil column above said traveling valve, the turns ratio of the primary and secondary windings of said first transformer being the ratio betweenv the cross-sectional area of said rod and the cross-sectional area ofsaid oil column, the turns ratio of the. primary and secondary windings of said second transformer being the ratio between the sum of the crosssectional areas of said sucker rod and oil column and the cross-sectional area of said oil column; and means for preventing current flow through said transformers. during periods representing the upstroke of said pump.

14. The combination in accordance with claim 13 wherein said means comprises a first rectifier connected in series with said first-mentioned voltage. source, said first-mentioned-inductance 

